If it's an army consisting of only archers + trebs, make capped rams/siege rams and fill it with infantry/elite skirmishers. The rams target the trebuchets, and if it gets destroyed, the skirmishers will do some damage. Another counter is of course trebuchets. If you repair them, you might manage to defend against the enemy trebuchets.

If you can pick a civ from the start with the sole purpose of destroying trebs defended by insane amounts of archers (perhaps Briton ones), I would recon that Mayan fully upgraded eagle warriors would not only manage to destroy the trebs, but also kill off the archers.

Let me be more specific than the only comment above worth upvoting, the problem with massing archers and trebuchets is that it will cost him an insane amount of gold. The counter for archers is elite skirmishers (and onagers for e skirms) and the counter for trebuchets is light cavalry. The best part is that neither of these units costs gold. So you will not be absolutely crushing his force, but while your castles hold it up, you will be able to pour troops at him without worrying about the cost, while he will have to switch his play style to include more halberdiers. Then you train a few onagers, which will utterly demolish his troops and leave the trebs defenceless. Happy hunting.

Rams total trebuchets. Trebs vs Trebs is okay if your opponent is stupid and doesn't have castles nearby. Honestly though... sometimes it's better to let him treb down a few buildings instead of throwing wave after wave of expensive cavalry/siege into an archer meatgrinder.

Yeah, sometimes it's better to just let him have a gold heavy army sit outside your base for a few minutes damaging buildings that were built for defense anyways. While they're tied down there, send your army out to mess them up. It'll be a lot of work and micro to defend and attack and you can exploit the inevitable slip ups that occur.

Not as easy countered as you make it seem to be. To counter war elephants with monks, you have to know they're coming to be able to make monks. You usually know this when they attack your base, but not so when they are in your opponent's base that you are attacking. A very real issue with them is that with only a handful of trebuchets, your castles will be down and the trebuchets gone long before the war elephants arrive.

If the defending guy already got fully upgraded elephants, you would've noticed and made monks a long time ago. If not, some halberdiers mixed in with the already massed archers will fend off the elephants. It's also a question of cost efficiency. Yes, you can spam elephants towards the trebs, but it's costly (five elephants is 1000 food i think).